Abstract
The toric ideals of 3×3 transportation polytopes \(\mathsf{T}_{\mathbf{rc}}\) are quadratically generated. The only exception is the Birkhoff polytope B 3.
If \(\mathsf{T}_{\mathbf{rc}}\) is not a multiple of B 3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3×3 transportation polytopes.
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Baldoni-Silva, W., de Loera, J., Vergne, M.: Counting integer flows in networks. Found. Comput. Math. 4(3), 277–314 (2004)
Beck, M., Chen, B., Fukshansky, L., Haase, C., Knutson, A., Reznick, B., Robins, S., Schürmann, A.: Problems from the cottonwood room. In: Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization. Contemp. Math., vol. 374, pp. 179–191. Am. Math. Soc., Providence (2005)
Bögvad, R.: On the homogeneous ideal of a projective nonsingular toric variety. arXiv:alg-geom/9501012 (1995)
Bruns, W., Gubeladze, J., Trung, N.: Normal polytopes,triangulations and Koszul algebras. J. Reine Angew. Math. 485, 123–160 (1997)
Diaconis, P., Eriksson, N.: Markov bases for noncommutative Fourier analysis of ranked data. J. Symb. Comput. 41(2), 182–195 (2006)
Ewald, G., Schmeinck, A.: Representation of the Hirzebruch-Kleinschmidt varieties by quadrics. Beitr. Algebra Geom. 34(2), 151–156 (1991)
Fulton, W.: Introduction to toric varieties. In: The William H. Roever Lectures in Geometry. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)
Gawrilow, E., Joswig, M.: Geometric reasoning with polymake. arXiv:math.CO/0507273 (2005)
Hellus, M., Hoa, L., Stückrad, J.: Gröbner bases for simplicial toric ideals. arXiv:0710.5347 (2007)
Koelman, R.: A criterion for the ideal of a projectively embedded surface to be generated by quadrics. Beitr. Algebra Geom. 34(1), 57–62 (1993)
Lee, C.W.: Subdivisions and triangulations of polytopes. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 271–290. CRC Press, New York (1997)
Lenz, M.: Toric ideals of flow polytopes. Master’s thesis, Freie Universität Berlin (2007). arXiv:0709.3570, see also arXiv:0801.0495
Ohsugi, H., Hibi, T.: Convex polytopes all of whose reverse lexicographic initial ideals are squarefree. Proc. Am. Math. Soc. 129(9), 2541–2546 (2001)
Piechnik, L.C.: Smooth reflexive 4-polytopes have quadratic triangulations. Undergraduate thesis, Duke University (2004)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)
Sturmfels, B.: Equations defining toric varieties. In: Algebraic Geometry—Santa Cruz 1995. Proc. Sympos. Pure Math., vol. 62, pp. 437–449. Am. Math. Soc., Providence (1997). arXiv:alg-geom/9610018
Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. Am. Math. Soc., Providence (1996)
Sullivant, S.: Compressed polytopes and statistical disclosure limitation. Tohôku Math. J. 58(3), 433–445 (2006). arXiv:math.CO/0412535
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Both authors were supported by Emmy Noether grant HA 4383/1 of the German Research Foundation (DFG).
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Haase, C., Paffenholz, A. Quadratic Gröbner bases for smooth 3×3 transportation polytopes. J Algebr Comb 30, 477–489 (2009). https://doi.org/10.1007/s10801-009-0173-4
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DOI: https://doi.org/10.1007/s10801-009-0173-4